Idealizer

In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal.^[1] Such an idealizer is given by

${\displaystyle \mathbb {I} _{S}(T)=\{s\in S\mid sT\subseteq T{\text{ and }}Ts\subseteq T\}.}$

In ring theory, if A is an additive subgroup of a ring R, then ${\displaystyle \mathbb {I} _{R}(A)}$ (defined in the multiplicative semigroup of R) is the largest subring of R in which A is a two-sided ideal.^[2][3]

In Lie algebra, if L is a Lie ring (or Lie algebra) with Lie product [x,y], and S is an additive subgroup of L, then the set

${\displaystyle \{r\in L\mid [r,S]\subseteq S\}}$

is classically called the normalizer of S, however it is apparent that this set is actually the Lie ring equivalent of the idealizer. It is not necessary to specify that [S,r] ⊆ S, because anticommutativity of the Lie product causes [s,r] = −[r,s] ∈ S. The Lie "normalizer" of S is the largest subring of L in which S is a Lie ideal.